Theorem cdlemksv 36132

Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013.)

Hypotheses

Ref	Expression
cdlemk.b	|- B = ( Base ` K )
cdlemk.l	|- .<_ = ( le ` K )
cdlemk.j	|- .\/ = ( join ` K )
cdlemk.a	|- A = ( Atoms ` K )
cdlemk.h	|- H = ( LHyp ` K )
cdlemk.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk.r	|- R = ( ( trL ` K ) ` W )
cdlemk.m	|- ./\ = ( meet ` K )
cdlemk.s	|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )

Assertion

Ref	Expression
cdlemksv	|- ( G e. T -> ( S ` G ) = ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )

Distinct variable groups:   ./\ , f   .\/ , f   f, F   f, i, G   f, N   P, f   R, f   T, f   f, W

Allowed substitution hints:   A( f, i)   B( f, i)   P( i)   R( i)   S( f, i)   T( i)   F( i)   H( f, i)   .\/ ( i)   K( f, i)   .<_ ( f, i)   ./\ ( i)   N( i)   W( i)

Proof of Theorem cdlemksv

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( f = G -> ( R ` f ) = ( R ` G ) )
2	1	oveq2d 6666	. . . . 5 |- ( f = G -> ( P .\/ ( R ` f ) ) = ( P .\/ ( R ` G ) ) )
3		coeq1 5279	. . . . . . 7 |- ( f = G -> ( f o. `' F ) = ( G o. `' F ) )
4	3	fveq2d 6195	. . . . . 6 |- ( f = G -> ( R ` ( f o. `' F ) ) = ( R ` ( G o. `' F ) ) )
5	4	oveq2d 6666	. . . . 5 |- ( f = G -> ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) = ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) )
6	2, 5	oveq12d 6668	. . . 4 |- ( f = G -> ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) )
7	6	eqeq2d 2632	. . 3 |- ( f = G -> ( ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) <-> ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )
8	7	riotabidv 6613	. 2 |- ( f = G -> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) = ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )
9		cdlemk.s	. 2 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10		riotaex 6615	. 2 |- ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) e. _V
11	8, 9, 10	fvmpt 6282	1 |- ( G e. T -> ( S ` G ) = ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( N ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   |-> cmpt 4729   `'ccnv 5113   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Atomscatm 34550   LHypclh 35270   LTrncltrn 35387   trLctrl 35445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653

This theorem is referenced by:  cdlemksel  36133  cdlemksv2  36135  cdlemkuvN  36152  cdlemkuel  36153  cdlemkuv2  36155  cdlemkuv-2N  36171  cdlemkuu  36183